{"title":"Probabilistic perturbation bounds of matrix decompositions","authors":"Petko H. Petkov","doi":"10.1002/nla.2582","DOIUrl":null,"url":null,"abstract":"In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Linear Algebra with Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/nla.2582","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems.
在本文中,我们根据马尔科夫不等式确定了随机扰动矩阵项的概率近似值。我们利用这些近似值,以规定概率推导出一些正交和单元矩阵分解的渐近分量扰动边界。我们证明,概率渐近界值的保守性明显低于相应的确定性扰动界值。作为案例研究,我们考虑用一种统一的方法来确定矩阵的 QR 分解、奇异值分解和舒尔分解的概率扰动边界,并对这些分解进行渐近分量扰动分析。结果表明,正交变换、奇异值和特征值的概率边界比相应的确定性渐近边界要严格得多。推导出的概率边界适用于大阶矩阵问题的扰动分析。
期刊介绍:
Manuscripts submitted to Numerical Linear Algebra with Applications should include large-scale broad-interest applications in which challenging computational results are integral to the approach investigated and analysed. Manuscripts that, in the Editor’s view, do not satisfy these conditions will not be accepted for review.
Numerical Linear Algebra with Applications receives submissions in areas that address developing, analysing and applying linear algebra algorithms for solving problems arising in multilinear (tensor) algebra, in statistics, such as Markov Chains, as well as in deterministic and stochastic modelling of large-scale networks, algorithm development, performance analysis or related computational aspects.
Topics covered include: Standard and Generalized Conjugate Gradients, Multigrid and Other Iterative Methods; Preconditioning Methods; Direct Solution Methods; Numerical Methods for Eigenproblems; Newton-like Methods for Nonlinear Equations; Parallel and Vectorizable Algorithms in Numerical Linear Algebra; Application of Methods of Numerical Linear Algebra in Science, Engineering and Economics.